Find minimum steps required to reach the end of a matrix | Set – 1

Given a 2D matrix consisting of positive integers, the task is to find the minimum number of steps required to reach the end(leftmost-bottom cell) of the matrix. If we are at cell (i, j) we can go to cells (i, j+arr[i][j]) or (i+arr[i][j], j). We can not go out of bounds. If no path exists, print -1.

Examples: 

Input : 
mat[][] = {{2, 1, 2},
           {1, 1, 1},
           {1, 1, 1}}
Output : 2
Explanation : The path will be {0, 0} -> {0, 2} -> {2, 2}
Thus, we are reaching end in two steps.

Input :
mat[][] = {{1, 1, 2},
           {1, 1, 1},
           {2, 1, 1}}
Output : 3

A simple solution is to explore all possible solutions. This will take exponential time.
Better approach: We can use dynamic programming to solve this problem in polynomial time.
Let’s decide the states of ‘dp’. We will build up our solution on 2d DP. 
Let’s say we are at cell {i, j}. We will try to find the minimum number of steps required to reach the cell (n-1, n-1) from this cell. 
We only have two possible paths i.e. to cells {i, j+arr[i][j]} or {i+arr[i][j], j}.
A simple recurrence relation will be: 

dp[i][j] = 1 + min(dp[i+arr[i][j]][j], dp[i][j+arr[i][j]])

Below is the implementation of the above idea: 

2

Time Complexity: O(N²).
Auxiliary Space: O(N²)